Fly-half Russell, who has so often been the architect of England's downfall, fired wide of the left post in the final minute at Twickenham following a dramatic touch down from Duhan van der Merwe.
First-half tries from Ben White and Huw Jones, either side of a converted score from England's Tommy Freeman, had put Gregor Townsend's team en route for an unprecedented fifth consecutive win in this fixture on Saturday.
But the Scots paid a heavy price for failing to turn their early dominance into a greater lead as eight points from Marcus Smith and a long-range Fin Smith penalty turned the contest in the hosts' favour, prior to the sensational finale.
"Scotland played a lot in our half and we didn't exit the way we wanted but we did fight and scrap," England captain Maro Itoje said.
"There is loads for us to get better at but we are delighted to win."
A fortnight on from a thrilling 26-25 victory over France, Steve Borthwick's side overcame a dismal first-half display to keep alive their Six Nations title hopes by the finest of margins.
England were installed as heavy favourites by bookmakers but it was Scotland who benefited from a blistering start as Tom Jordan's burst down the left wing allowed scrum-half White to dive over.
The hosts produced a perfect response to the fourth-minute setback, with Freeman adjudged to have grounded a close-range finish at the end of sustained pressure before Marcus Smith slotted the conversion.
Van der Merwe and Blair Kinghorn, who helped assist the opening score, then combined to release try-scorer Jones on the left following a swift line-out move which began on the opposite flank.
Scotland's superiority should really have resulted in greater breathing space on the scoreboard but with Russell having also missed his initial two conversion attempts, they led only 10-7 at halftime.
Borthwick wasted little time in turning to his bench in a bid to shift momentum as he introduced experienced duo Jamie George and Elliot Daly inside five minutes of the restart.
A Smith penalty soon levelled proceedings, and England led for the first time in the 67th minute when he slotted another from in front of the posts.
England fly-half Fin Smith, who starred against France, nervelessly split the posts from halfway to give the home side a 16-10 advantage nine minutes from time.
But the game was back in the balance when Scottish replacement Stafford McDowall powered through on a break, culminating in a van der Merwe score on the left.
Russell was left with a match-winning moment but his effort sailed wide before England survived the frantic finish.
Man-of-the-match van der Merwe said Scotland had failed to take opportunities in the first half.
"Fair play to England's defence, in the first half they kept us out," he told ITV.
"Coming into the second half we had belief that we could win this game. We didn't have enough ball in the second half. England were applying pressure at the breakdown and they got the ball."
34°C